Lagrange Error Bound Worksheet
A Grade 12 Calculus worksheet focusing on understanding and applying the Lagrange Error Bound for Taylor polynomials.
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Lagrange Error Bound Worksheet
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Read each question carefully and show all your work. Round your answers to three decimal places where appropriate.
1. State the formula for the Lagrange Error Bound (remainder) for a Taylor polynomial P_n(x) approximating a function f(x) centered at x = c.
2. The Lagrange Error Bound provides an for the maximum possible error when using a Taylor polynomial to approximate a function.
3. To find the maximum value of the (n+1)th derivative, M, we often need to analyze the function's behavior on the between the center of the Taylor series and the point of approximation.
4. If a Taylor polynomial of degree n is used to approximate a function f(x), the Lagrange Error Bound involves the (n+1)th derivative of f(x). What does this derivative represent?
The exact value of the function.
The error in the approximation.
The rate of change of the (n)th derivative.
The remainder term.
5. Consider the function f(x) = e^x. Use the 3rd degree Taylor polynomial centered at x = 0 to approximate e^(0.5). Calculate the maximum possible error using the Lagrange Error Bound.
Hint: Find the 4th derivative of f(x) and its maximum value on the interval [0, 0.5].
6. The Lagrange Error Bound always gives the exact error of a Taylor polynomial approximation.
True
False